What is the integral from 0 to 4 of x(1+x/2) ?

The integral from 0 to 4 of x(1+x/2) is 56/3

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integral from 0 to 4 of x(1+x/2)

\int_{0}^{4} x (1 + \frac{x}{2}) d x

\frac{56}{3}

Solution steps

\int_{0}^{4} x (1 + \frac{x}{2}) d x

Expand

x (1 + \frac{x}{2}) : x + \frac{x ^{2}}{2}

= \int_{0}^{4} x + \frac{x ^{2}}{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{4} x d x + \int_{0}^{4} \frac{x ^{2}}{2} d x

\int_{0}^{4} x d x = 8

\int_{0}^{4} \frac{x ^{2}}{2} d x = \frac{32}{3}

= 8 + \frac{32}{3}

Simplify

8 + \frac{32}{3} : \frac{56}{3}

= \frac{56}{3}

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What is the integral from 0 to 4 of x(1+x/2) ?
The integral from 0 to 4 of x(1+x/2) is 56/3